Theorem matval 20036

Description: Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)

Hypotheses

Ref	Expression
matval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
matval.g	⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
matval.t	⊢ · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)

Assertion

Ref	Expression
matval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Proof of Theorem matval

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		matval.a	. 2 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		elex 3185	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
4		id 22	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → 𝑛 = 𝑁)
5	4	sqxpeqd 5065	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 × 𝑛) = (𝑁 × 𝑁))
6	3, 5	oveqan12rd 6569	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = (𝑅 freeLMod (𝑁 × 𝑁)))
7		matval.g	. . . . . 6 ⊢ 𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))
8	6, 7	syl6eqr 2662	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 freeLMod (𝑛 × 𝑛)) = 𝐺)
9	4, 4, 4	oteq123d 4355	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ⟨𝑛, 𝑛, 𝑛⟩ = ⟨𝑁, 𝑁, 𝑁⟩)
10	3, 9	oveqan12rd 6569	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
11		matval.t	. . . . . . 7 ⊢ · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
12	10, 11	syl6eqr 2662	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩) = · )
13	12	opeq2d 4347	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩ = ⟨(.r‘ndx), · ⟩)
14	8, 13	oveq12d 6567	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
15		df-mat 20033	. . . 4 ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
16		ovex 6577	. . . 4 ⊢ (𝐺 sSet ⟨(.r‘ndx), · ⟩) ∈ V
17	14, 15, 16	ovmpt2a 6689	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
18	2, 17	sylan2 490	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
19	1, 18	syl5eq 2656	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133   × cxp 5036  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ndxcnx 15692   sSet csts 15693  ._rcmulr 15769   freeLMod cfrlm 19909   maMul cmmul 20008   Mat cmat 20032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-mat 20033

This theorem is referenced by:  matbas  20038  matplusg  20039  matsca  20040  matvsca  20041  matmulr  20063